On nonvanishing solutions of a class of functional equations (Q1880841)

Let \(\psi\) be a function holomorphic on a neighbourhood of the closed unit disk \(\mathbb{D}\subset\mathbb{C}\), such that \(\psi\) is nonnegative and increasing on \([0,1]\), \(\psi'(0)> 0\), \(\psi(1)< 1\), and \(|\varphi'(z)|\leq \psi'(| z|)\) in \(\mathbb{D}\). The author studies boundary values of holomorphic functions \(w\) in \(\mathbb{D}\) satisfying \(w(0)= 0\) and \[ \lim_{z\to\xi}| w'(z)|+ \psi(| w(z)|)= 1,\qquad \xi\in\partial\mathbb{D}. \] For example, it is shown that if \(w\) is analytic in a neighbourhood of \(\xi_0\in\partial\mathbb{D}\), then \(w(\xi_0)\neq 0\).